GUTHRIE CENTER, Iowa—Craig Barnum, a school administrator, took the afternoon off recently and drove three hours to Sen. Chuck Grassley’s town-hall meeting to press the Republican senator on the GOP plan to overhaul health care.

Mr. Barnum has health insurance through his work. But his family depends on Medicaid to help cover the medical costs of his 9-year-old son, Koan, whose delayed development means he needs expensive therapy and equipment including leg braces.